Rank-one convexity implies quasi-convexity on certain hypersurfaces

Nirmalendu Chaudhuri; Stefan Müller

doi:10.1017/s0308210500002912

Rank-one convexity implies quasi-convexity on certain hypersurfaces

Nirmalendu Chaudhuri^*, Stefan Müller

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

We show that, if f : double struck M sign^2×2 → ℝ is rank-one convex on the Hyperboloid H_D^- := {X ∈ S^2×2 : det X = -D, X₁₁ ≥ c > 0}, D ≥ 0, S^2×2 is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on double struck M sign^2×2 uniformly on compact subsets of H_D ^-. Equivalently, every gradient Young measure supported on a compact subset of H_D^- is a laminate.

Original language	English
Pages (from-to)	1263-1272
Number of pages	10
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume	133
Issue number	6
DOIs	https://doi.org/10.1017/s0308210500002912
Publication status	Published - 2003
Externally published	Yes

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abstract = "We show that, if f : double struck M sign2×2 → ℝ is rank-one convex on the Hyperboloid HD- := \{X ∈ S2×2 : det X = -D, X11 ≥ c > 0\}, D ≥ 0, S2×2 is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on double struck M sign2×2 uniformly on compact subsets of HD -. Equivalently, every gradient Young measure supported on a compact subset of HD- is a laminate.",

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N2 - We show that, if f : double struck M sign2×2 → ℝ is rank-one convex on the Hyperboloid HD- := {X ∈ S2×2 : det X = -D, X11 ≥ c > 0}, D ≥ 0, S2×2 is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on double struck M sign2×2 uniformly on compact subsets of HD -. Equivalently, every gradient Young measure supported on a compact subset of HD- is a laminate.

AB - We show that, if f : double struck M sign2×2 → ℝ is rank-one convex on the Hyperboloid HD- := {X ∈ S2×2 : det X = -D, X11 ≥ c > 0}, D ≥ 0, S2×2 is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on double struck M sign2×2 uniformly on compact subsets of HD -. Equivalently, every gradient Young measure supported on a compact subset of HD- is a laminate.

UR - https://www.scopus.com/pages/publications/1642532570

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DO - 10.1017/s0308210500002912

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JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

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ER -

Rank-one convexity implies quasi-convexity on certain hypersurfaces

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